Neat Epimorphisms of Modules

A homomorphism f : A → B of modules is said to be a neat homomorphism if it has no proper extension in the injective envelope E(A) of A. We show that they are exactly the homomorphisms that preserve the essential monomorphisms in pushout diagrams as observed by Zöschinger in the case of abelian groups. Over a commutative domain where every maximal ideal is invertible, if an epimorphism A → B of modules is neat and Rad A = 0, then we show that it is an isomorphism. We prove that a homomorphism f : A → B of modules over a Dedekind domain is neat if and only if Ker f ⊆ Rad A and Im f is closed in B (that is, Im fs has no proper essential extension in B). In that proof, we also use some relations between the related proper classes of short exact sequences of modules. We investigate which proper class defining properties are satisfied by the class of all short exact sequences of modules determined by neat epimorphisms. Unlike the class determined by neat monomorphisms, this class is not a proper class unless R is a semisimple ring.